P is x% more than Q. Q is (x - 10)% less than R. If P > R, what is the range…

Given relations:

• P is x% more than Q, so P = Q(1 + x/100).

• Q is (x - 10)% less than R, so Q = R(1 - (x-10)/100) = R((110 - x)/100).

Express R in terms of Q:

R = Q / ((110 - x)/100) = 100Q/(110 - x).

Given P > R, substitute the expressions and cancel Q (assuming Q > 0):

(100 + x)/100 > 100/(110 - x).

Cross-multiply to remove denominators (valid if 110 - x > 0; we'll consider this):

(100 + x)(110 - x) > 10000.

Expand and rearrange:

• 11000 + 10x - x^2 > 10000 → 1000 + 10x - x^2 > 0

• Rewrite: x^2 - 10x - 1000 < 0.

Solve the quadratic equation x^2 - 10x - 1000 = 0 to find the boundary points:

Roots: x = [10 ± sqrt(100 + 4000)]/2 = [10 ± sqrt(4100)]/2 ≈ -27.02 and 37.02.

Because the quadratic opens upward, the inequality x^2 - 10x - 1000 < 0 holds between the roots:

-27.02 < x < 37.02.

Interpretation in the percentage context:

• The phrase "Q is (x - 10)% less than R" implies x - 10 ≥ 0 (so that it is actually a reduction), hence x ≥ 10.

• Combine this with the inequality: 10 ≤ x < 37.02.

Therefore, in percentage terms x can range from 10% up to about 37%. (So the correct choice giving 10% to 37% is justified.)